Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
515 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 515 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,31)(24,30)(25,27)(26,29)(28,33)(32,36)(34,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,27)(23,32)(29,30)(33,34);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23,25,32)(22,28,27,35)(24,33,26,34)(29,31,30,36);(6,12)(10,13)(11,20)(18,19)(21,24,25,26)(22,29,27,30)(23,33,32,34)(28,31,35,36)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107928, #108024, #110212, … |
| maximal normal subgroup of | #107928, #108024, #110212, … |
| maximal quotient of | #215684, #215688, #215692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14825);